Fixed points and stability of A class of Stochastic dynamical system driven by Brownian motion

In real life, many models and systems are affected by random phenomena. For this reason, experts and scholars propose to describe these stochastic processes with Brownian motion respectively. In this paper we consider a kind of stochastic Vollterra dynamical systems of nonconvolution type and give some new conditions to ensure that the zero solution is asymptotically stable in mean square by means of fixed point method. The theorems of asymptotically stability in mean square with a necessary conditions are proved. Some results of related papers are improved.


Introduction
The Lyapunov direct method is very effective in establishing stability results for various dynamical systems. However, there are a host of problems which can not be solved. Recently, Burton and other authors have studied stability [1][2][3][4][5][6][7][8] by using the fixed point theory. Research results show that many of the problems can be solved by the fixed point theory when we use Lyapunov direct method in stability studies. The Lyapunov direct method usually requires a point state condition, while the stability result of the fixed point theory requires an average nature condition. Following it, Zhang [9] also use the fixed point methods to study the stability.
When we study the stability of stochastic dynamical systems, some methods were applied [10][11] . Recently, Luo [12] firstly used fixed point method to study the stability of stochastic dynamical systems. However, while we study the stability by means of fixed point method, the operator we define is very important. In this paper, we consider the asymptotically stability in mean square of a kind of stochastic dynamical systems of nonconvolution type by means of fixed point method too.
According to the authors, one of the innovations of this paper is that few experts have used the fixed point method to study the destabilizing Volterra dynamical system of nonconvolution. In addition, the result of Burton [1] is improved and generalized by defining different operator. It is shown that the fixed point method is more flexible and efficient than other methods such as Lyapunov's direct method in the study of the null solution stability of the logistic volatile terra system of nonconvolution type.
The rest of this paper is arranged as following. First, we state the main theorems and relative proofs in section 1. Secondly, , an example shows that our stability results are indeed better than those in [1] in section 2. F contains all P-null sets and right continuous. Let {ω(t), t ≥ 0}be a standard one-dimension Wiener process defined on {Ω, F , P}. The mapping a(t) ∈ C(R + , R).

Main results
Consider the following one-dimensional stochastic Vollterra dynamical system of nonconvolution type and b : R + ×R + → R. Assume that there exists a unique solution in system (1) which is denoting by () xt .
The function g satisfy the Lipschitz condition and positive constant M and N exists, such that for Burton [1] has studied the asymptotically stability of this system. And have the result as following: Theorem A[Burton [1] ] Assume there exist 1   , such that when t ≥ 0,  (2) is mean square asymptotic stable at t0=0. We will try to study the asymptotically stability in mean square of system (1) by using fixed point method in section 2. The result of Burton [1] is improved and generalized by defining different operator.

Theorem 1
Assume there exist a continuous function h(t) : [0, ∞) → R + and η(0, 1), such that when t ≥ 0, Firstly, it is obvious that  is continuous. Secondly, we prove  (S)  S, that is Thus we get that Ψ is contractive by (6). According to the contraction mapping principle, Ψ has a unique fixed point () In order to acquire the mean square asymptotic stability, we should prove that it is mean square stable in the zero solution of system (1 (3) and (6) that